Some local (at p) properties of residual Galois representations

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1 Some local (at ) roerties of residual Galois reresentations Johnson Jia, Krzysztof Klosin March 5, Preliminary results In this talk we are going to discuss some local roerties of (mod ) Galois reresentations at the rime. Let be a rime number, N a ositive integer rime to, k an integer and ɛ : (Z/NZ) F a character. Let f be a cus form of tye (N, k, ɛ). We also assume that f is an eigenform for all the Hecke oerators T l (any rime l) with eigenvalue a l F. For the definition of the Hecke algebra we refer to Kirsten s talk. We will denote by G Q the absolute Galois grou of Q, and by χ the mod cyclotomic character. We work with arithmetic Frobenius elements. Theorem 1.1 (Deligne). There exists a unique (u to isomorhism) continuous semi-simle reresentation ρ f : G Q GL(V ), where V is a two-dimensional F -vector sace, such that for all rimes l N, ρ f is unramified at l, tr ρ f (Frob l ) = a l det ρ f (Frob l ) = ɛ(l)l k 1. Remark 1.2. By choosing a basis of V we can treat ρ f as having image in GL 2 (F ). It then follows from continuity of ρ f that there exists a finite extension κ of F such that ρ f factors through some ma G Q GL 2 (κ) GL 2 (F ) taht we may also denote ρ f. We fix an embedding Q Q. This determines a rime of the integral closure Z of Z in Q over. Let G G Q denote the decomosition grou of. Our goal is to study ρ f, := ρ f G. We will denote by I the inertia grou inside G, by I w the wild inertia subgrou, and set I t = I /I w to be the tame inertia quotient. We will identify G with Gal(Q /Q ) and G /I with Gal(F /F ). Moreover, for any a F we denote by λ a the unramified character φ : G F such that φ(frob ) = a. Theorem 1.3 (Deligne). Let f be a cus form of tye (N, k, ɛ) with 2 k + 1. Assume f is a Hecke eigenform with eigenvalues a l, and that a 0. Then ρ f, is reducible and [ ] ρ f, χ k 1 λ = ɛ()/a. Remark 1.4. Theorem 1.3 was roved by Deligne in a letter to Serre using -adic etale cohomology and mod vanishing cycles calculations on modular curves in characteristic ; in articular his roof uses his construction of -adic Galois reresentations associated to eigenforms. Proosition 1.5. Let f be a cus form of tye (N, 1, ɛ). Assume that f is a Hecke eigenform and T f = a f. Suose that the olynomial P (X) := X 2 a X + ɛ() (1) has two distinct roots in F, or that f is the reduction of a characteristic zero eigenform of weight 1 and level N. Then ρ f, is unramified. λ a 1

2 Proof. We omit the roof of the case when f is the reduction of a characteristic zero eigenform (cf. Theorem 4.1 in [DS74]), so assume P (X) has distinct roots in F. Let f(q) = b n q n be the q-exansion of f at some cus. In his talk Hui discussed the action of the oerator V on q-exansions of cus forms. In articular V f is a cus form of tye (N,, ɛ) and V f(q) = b n q n. On the other hand, Bryden defined the Hasse invariant A, which is a modular form of tye (1, 1, 1) and has q-exansion 1 at all cuss. Hence Af is a modular form of tye (N,, ɛ) and Af has q-exansion b n q n. As discussed in Hui s talk, both Af and V f are eigenforms for all T l with l and their T l -eigenvalues are the same as those of f. On the other hand, since T f = a f and f = ɛ()f, it follows from general identities that and T (Af) = a Af ɛ()v f (2) T (V f) = Af. (3) Let W be the two-dimensional sace sanned by Af and V f. It follows from formulas (2) and (3) that P (X) is the characteristic olynomial for the action of T on W. Let α α be the roots of P (X). Let g, g be the corresonding normalized eigenforms of tye (N,, ɛ). Since f, g and g have the same eigenvalues for T l, l, we have tr ρ f = tr ρ g = tr ρ g by the Tchebotarev density theorem. We also have det ρ f = det ρ g = det ρ g = ɛ. Hence the semi-simle reresentations ρ f, ρ g and ρ g are isomorhic by the Brauer-Nesbitt theorem. Since αα = ɛ() 0, both α and α are non-zero. Hence we can aly Theorem 1.3 and conclude that we have [ ] [ ] ρ g, λɛ()/α λα = = (4) λ α λ α and ρ g, = [ ] [ ] λɛ()/α λα =. (5) λ α λ α Since α α, the reresentation sace ρ f, contains lines with distinct characters λ α and λ α, so ρ f, = λ α λ α is unramified. Theorem 1.6 (Gross). Let f = a n q n be as in Theorem 1.3, and suose that a 2 ɛ() if k =. Then ρ f, I w is trivial if and only if there exists a cusidal eigenform f = a nq n of tye (N, k, ɛ), where k = + 1 k, such that la l = l k a l (6) for all l. Proof. We are only going to rove that the existence of f imlies that ρ f, is tamely ramified. For the converse (which is the hard art of this theorem), see Theorem in [Gro90]. Let f be as in the statement of the theorem and let a l denote the eigenvalue of T l corresonding to f. Formula (6) imlies that tr ρ f (Frob l ) = tr (ρ f χ k 1 )(Frob l ) for all l N, where χ denotes the mod cyclotomic character. Hence by the Tchebotarev density theorem together with the Brauer-Nesbitt theorem as before, we get ρ f = ρf χ k 1. (7) First assume k. We begin by show that a 0. Note that for all rimes l, we have a l = lk 1 a l by assumtion. If a = 0, we can also write a = k 1 a as k 1. Hence f of weight k and θ k 1 f of weight k have the same q-exansions. Here θ is the oerator whose roerties were discussed in Kirsten s talk. Hence θ k 1 f = A k f, which means that θ k 1 f has filtration k. However, it was roved in Kirsten s talk that θ k 1 f has filtration k, since f has weight at most 1, which yields a contradiction. Hence a 0 if k. Alying Theorem 1.3 to the reresentation ρ f (still assuming k ), we see that there is a line L f in the sace V of ρ f such that G acts on L f via the character χ k 1 λ ɛ()/a. On the other hand alying Theorem 1.3 to the reresentation ρ f, we get a line L f in its reresentation sace on which G acts via the character χ k 1 λ ɛ()/a and hence a line L f V on which G acts via the character χ k 1 (χ k 1 λ ɛ()/a ). 2

3 In articular, the action of I on L f is via χ k 1+k 1 = χ 1 = 1 and on L f via χ k 1. Since k by assumtion, so χ k 1, L f L f and so we conclude that V = L f L f. Thus ρ f, and hence also ρ f, is diagonalizable; i.e., ρ f (σ) has order rime to for every σ G. Thus ρ f I w is trivial. Now assume k =. Then f = a nq n is of weight 1. Note that f is in the san W of V f and Af. Indeed, by looking at q-exansions, we see that θ((a a )V f + Af f) = 0, and so there exists a form h = c n q n of weight 1 such that (a a )V f + Af f = V h, and h is an eigenform for T l with eigenvalue a l if l since V commutes with such T l in weights 1 and (and V is injective). Moreover, the -th coefficient of the q-exansion of V h is zero, so c 1 = 0. Since c l = a l c 1 = 0 for all l, we have θ(h) = 0. As θ is injective on forms of weight rime to, we get h = 0, and so we conclude that f W, as claimed. As a ɛ()/a by assumtion, and the characteristic olynomial for T on W has constant term ɛ() (by matrix calculation) with a as a root (since f W ), there is a unique Hecke eigenform g W whose eigenvalue for T is b := ɛ()/a. Then by alying Theorem 1.3 to ρ f and ρ g = ρf, we get a line L f in the sace W of ρ f on which G acts via the unramified character λ ɛ()/a and a line L g on which G acts via λ ɛ()/b. Since a 2 ɛ(), we again get V = L f L g and conclude that ρ f I w is trivial. Note that in this case ρ f, is in fact unramified. 2 A theorem of Mazur Let be an odd rime and l a rime with l 1 mod. (We allow l =.) Let f be a normalized cusidal eigenform of weight 2, level N, and trivial charater. We dro the assumtion that N. Suose N = Ml with l M. Here is a basic level lowering theorem. Theorem 2.1 (Mazur). Suose the Galois reresentation ρ : G Q GL 2 (F ) attached to f is irreducible. If l, then assume that ρ Gl is unramified, and if l =, then assume that ρ is finite at. Then ρ is modular of tye (M, 2, 1). Proof. We will suress some of the technical details. Let X 0 (N) over Z l be the coarse moduli sace for the Γ 0 (N)-moduli roblem on generalized ellitic curves over Z (l) -schemes. Let J 0 (N) be the jacobian of X 0 (N) over Q. Let J 0 (N) be the Neron model of J 0 (N) over Z (l). It follows from the work of Deligne- Raoort that X 0 (N) Fl is the union of two coies of X 0 (M) Fl, which intersect transversally at geometric oints corresonding to recisely suersingular ellitic curves. Their theory shows that X 0 (M) Fl is smooth, as l M. Lemma 2.2. The normalization ma X 0 (M) Fl X 0 (M) Fl X 0 (N) Fl induces a short exact sequence 0 T 0 (N) Fl J 0 (N) 0 F l J 0 (M) Fl J 0 (M) Fl 0, (8) where T 0 (N) Fl is the torus whose character grou X := Hom Fl (T 0 (N) Fl, (G m ) Fl ) as a Z[Gal(F l /F l )]- module is the grou of degree zero divisors on X 0 (M) Fl with suort in the suersingular oints. Proof. The existence of exact sequence (8) is a general fact about semi-stable curves, cf. Examle 8 in section 9.2 of [BLR90]. To any semi-stable curve Y over an algebraically closed field one associates the grah Γ(Y ) whose vertices are the irreducible comonents of Y and whose edges are the singular oints of Y : an edge corresonding to a oint P joins vertices V 1 and V 2 if and only if the irreducible comonents corresonding to V 1 and V 2 intersect at P. (We get a loo at a vertex if it lies on only one irreducible comonent, corresonding to formal self-crossing.) The character grou X can be canonically identified with H 1 (Γ(X 0 (N) Fl ), Z) as a Z[Gal(F l /F l )]-module. Using this identification, one roves that X has the alternative descrition in terms of the suersingular geometric oints. There is yet another tautological short exact sequence 0 J 0 (N) 0 F l J 0 (N) Fl Φ 0 (N) Fl 0, (9) 3

4 where Φ 0 (N) Fl is the finite etale grou scheme of comonents of J 0 (N) Fl. The Hecke algebra T of level N acts on J 0 (N) and by the Neron maing roerty also on J 0 (N). This action resects the exact sequences (8) and (9) due to functoriality considerations with identity comonents and maximal toric arts. Since ρ is absolutely irreducible and modular of tye (N, 2, 1), by the Brauer-Nesbitt theorem and the triviality of Brauer grous of finite fields there exists a maximal ideal m of T with an embedding κ := T/m F and a two-dimensional κ-vector sace V with G Q -action such that the induced action on F κ V is via ρ and such that J 0 (N)(Q)[m] is isomorhic to the direct sum of a finite number of coies of V as κ[g Q ]-modules. First assume that l =, as this is the case that will be of interest to us. Let W be the finite etale κ-vector sace scheme over Q such that W (Q) = V as κ[g Q ]-modules. By assumtion ρ is finite at, so (by oddness of and Raynaud s theorems when e < 1 ) there exists a finite flat κ-vector sace scheme W over Z (l) extending W. By the remarks above we can choose an injection of V into J 0 (N)(Q) which gives rise to a closed immersion of W into J 0 (N). Lemma 2.3. There exists a closed immersion W J 0 (N) rolonging the closed immersion W J 0 (N). Proof. We are going to rove the following more general statement: Claim 2.4. Let R denote a comlete dvr of mixed characteristic (0, ) with absolute ramification index e < 1. Denote by K the fraction field of R. Let G be a finite flat commutative -grou over R, and A an abelian variety over K with semi-stable reduction. Let A be the Neron model (over R) of A (so A has closed fiber with semi-abelian identity comonent, whence A[n] is quasi-finite, flat, and searated over R for every nonzero n in Z). Suose there exists a closed immersion ι K : G A, where G = G K. Furthermore, assume that the Galois module (A[]/A[] f )(K) is unramified, where A[] f denotes the finite art of the quasi-finite flat searated R-grou A[]. Then ι K extends to a closed immersion ι : G A. We remark here that the hyotheses of semi-stable reduction and unramifiedness are satisfied by A = J 0 (N) over Z (l). This is a consequence of [Gro72] and uses the fact that X 0 (N) has semi-stable reduction over Z (l) (as l M). Proof of the Claim. We first show that ι K G 0 K extends. The comosite G 0 K G A[ n ] K (A[ n ]/A[ n ] f ) K is the zero ma since, as discussed by Nick in his talk, Raynaud s results on finite flat grou schemes imly that a finite flat connected R-grou has no nonzero unramified quotients of its K-fiber. Hence we have a commutative diagram G A[] K G 0 K (A[] f ) K where all the mas are closed immersions. Now, using the fact that A[] f is finite flat (as oosed to A[], which is generally only quasi-finite and flat), the lower horizontal arrow extends to a closed immersion G 0 A[] f by Raynaud s Theorem ([Ray74], Corollary 3.3.6). As A[] 0 f A 0, we have roved that ι K G 0 K extends. We will now show that ι extends. Since G 0 is a finite flat commutative grou, it makes sense to talk about the quotient (G A 0 )/G 0 with G 0 embedded by the twisted diagonal x (x, x). We have a commutative diagram G (G A 0 )/G 0. (10) G 0 A 0 4

5 Our goal is to show that the closed embedding G 0 A factors through the left vertical arrow in diagram (10). First note that it suffices to show that (G A 0 )/G 0 is smooth. Indeed, the natural addition ma G A A factors through (G A)/(G 0 ) K A, so if (G A 0 )/G 0 is smooth, then (G A)/(G 0 ) K A extends to φ : (G A 0 )/G 0 A by the Neron maing roerty. Diagram (10) gets amalgamated to a diagram G (G A 0 )/G 0 φ. (11) G 0 A 0 To establish commutativity of diagram (11) it is enough (by searatedness and flatness over R) to check commutativity of the right triangle on generic fibers. Since (A 0 ) K = A, the corresonding triangle on generic fibers clearly commutes, so we get the desired result concerning (10). It remains to rove that (G A 0 )/G 0 is smooth over R. Note that G A 0 is of finite tye and flat as are G and A 0. As G 0 is finite and flat over R, the ma G A 0 (G A 0 )/G 0 is finite, flat and surjective. Hence (G A 0 )/G 0 is of finite tye and flat over R. Thus to verify smoothness of (G A 0 )/G 0, it is enough to check smoothness of fibers. As the geometric generic fiber is clearly smooth (it is just a roduct of coies of A), it remains to check smoothness of the closed fiber. This follows from the following Fact. Fact 2.5. Let k be a field, G a finite commutative k-grou and H a smooth commutative k-grou. Suose we are given a closed immersion of k grous G 0 H. Then (G H)/G 0 is k-smooth, where G 0 G H is g 0 (g 0, (g 0 ) 1 ). To rove Fact 2.5, we may assume that k is algebraically closed. Then G = G 0 G et, hence (G H)/G 0 = G et H, which is smooth. Lemma 2.6. The image of W F in Φ 0 (N) F is zero. A Proof. It is a theorem of Ribet (cf. Theorem 3.12 in [Rib90]) that Φ 0 (N) F is Eisenstein in the sense that for all q rime to, the oerator T q acts on it via multilication by q + 1. Thus the action of T on Φ 0 (N) F factors through T/I, where I is the ideal of T generated by the elements T q q 1 for q rime to. Note that since ρ is irreducible, I is relatively rime to m. Indeed, if m and I were not relatively rime, then T q q + 1 mod m for almost all q. By Tchebotarev Denisty Theorem we have then tr ρ = 1 + χ and det ρ = χ. Brauer-Nesbitt Theorem imlies then that ρ ss = [ 1 χ ], which contradicts the irreducibility of ρ. The Lemma follows. Lemma 2.6 imlies that W F lands inside J 0 (N) 0 F. From now on assume that ρ is not modular of tye (M, 2, 1). Then it follows (since ρ is irreducible) that the image of W F inside J 0 (M) F J 0 (M) F must be trivial ([Rib90], Thoerem 3.11). Hence, W F lands inside the torus T 0 (N) F. The Hecke algebra acts on the character grou X, and Ribet showed in [Rib90] (Lemma 6.3) that V Hom(X/mX, µ ), where µ denotes the Galois modules of -th roots of 1 over Q. Lemma 2.7. The action of Frob on X/mX coincides with the action of T and of w, where w denotes the level Atkin-Lehner involution. Proof. For the roof that the actions of T and of w on T 0 (N) F (and hence on X/mX) coincide, see [Rib90], Proosition 3.7. We will show that the action of Frob and of w coincide on X/mX. Since X = H 1 (Γ(X 0 (N) F ), Z), it is enough to consider the action of Frob and of w on Γ(X 0 (N) F ). By using the moduli interretation of X 0 (N) F one sees that Frob and w are both involutions and have the same effect on the edges of Γ(X 0 (N) F ), but Frob fixes the vertices of Γ(X 0 (N) F ), while w swas them. Hence Frob w swas the vertices of Γ(X 0 (N) F ) and fixes its edges, which on the homology has the same effect as multilication by -1. 5

6 Since w is an involution and κ is a field, Lemma 2.7 imlies that G acts on Hom(X/mX, µ ) via ψχ, where ψ is an unramified quadratic character. In articular the action of G on det κ V is via ψ 2 χ 2 = χ 2. On the other hand, G acts on det κ V by det ρ = χ, hence χ = χ 2, and thus χ must be trivial, which is an obvious contradiction since is odd. This finishes the roof of Theorem 2.1 in the case when l =. The case l roceeds along the same lines, but is simler as one does not need an analogue of Lemma 2.3. Instead we note that when l, the Galois module V is unramified by assumtion. Thus there exists a finite etale κ-vector sace scheme W over Z (l) such that V = W(Q) as κ[g Q ]-modules. Hence the injection V J 0 (N)(Q)[m] gives rise to the a closed immersion W Q J 0 (N). As l, W is smooth over Z (l), hence the extension W J 0 (N) exists by the Neron maing roerty. We claim that it is a closed immerion. Indeed, let W denote the schematic closure of the image of W in J 0 (N). Then W is finite and flat over R, but since it is an l-grou with l, it is also etale. Hence the fact that W Q WQ imlies that W W. In that case the final conclusion that χ = χ 2 imlies that l 1 mod, which was assumed not to be the case. References [BLR90] Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud. Néron models, volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Sringer-Verlag, Berlin, [DS74] Pierre Deligne and Jean-Pierre Serre. Formes modulaires de oids 1. Ann. Sci. École Norm. Su. (4), 7: (1975), [Gro72] Alexandre Grothendieck. Sga7 i, exose ix. Lecture Notes in Mathematics, 288(2): , [Gro90] Benedict H. Gross. A tameness criterion for Galois reresentations associated to modular forms (mod ). Duke Math. J., 61(2): , [Ray74] Michel Raynaud. Schémas en groues de tye (,..., ). Bull. Soc. Math. France, 102: , [Rib90] K. A. Ribet. On modular reresentations of Gal(Q/Q) arising from modular forms. Invent. Math., 100(2): ,